On two ways to use determinantal point processes for Monte Carlo integration
For researchers in Monte Carlo integration, this work clarifies trade-offs between two DPP-based approaches, offering practical guidance and new sampling methods.
The paper compares two determinantal point process (DPP)-based Monte Carlo estimators: one with variance rate O(N^{-(1+1/d)}) for smooth functions but using a fixed DPP, and another unbiased estimator with rate O(1/N) but requiring a DPP tailored to the integrand. The authors generalize both to continuous settings and provide sampling algorithms.
The standard Monte Carlo estimator $\widehat{I}_N^{\mathrm{MC}}$ of $\int fdω$ relies on independent samples from $ω$ and has variance of order $1/N$. Replacing the samples with a determinantal point process (DPP), a repulsive distribution, makes the estimator consistent, with variance rates that depend on how the DPP is adapted to $f$ and $ω$. We examine two existing DPP-based estimators: one by Bardenet & Hardy (2020) with a rate of $\mathcal{O}(N^{-(1+1/d)})$ for smooth $f$, but relying on a fixed DPP. The other, by Ermakov & Zolotukhin (1960), is unbiased with rate of order $1/N$, like Monte Carlo, but its DPP is tailored to $f$. We revisit these estimators, generalize them to continuous settings, and provide sampling algorithms.